Order and Disorder in Liquid-Crystalline Elastomers - Liquid Crystal Elastomers: Materials and Applications - page 218

density of rigid crosslinks) and subsequently to a stretched Lorentzian (large
density of rigid crosslinks).
The signature of disorder is, according to the theories of random disorder dis-
cussed in Sect. 4.3 , a shape of the diffraction profile corresponding to a stretched
Gaussian with b

0.7. This value corresponds approximately to a square Lorentzian
[last term of ( 13 ) ]. For the flexible crosslinker V1, this point is reached for a crosslink
concentration of about 15%, which is definitely much larger than predicted theoreti-
cally. The stiff crosslinker V8 shows a crossover to disorder at smaller concentrations
than V1, providing a better connectivity with theory. Taking the onset of disorder for
V8 at 7-8% and a percolation threshold c 0

4%, we arrive at c c 0 being
approximately equal to 3-4%, in reasonable correspondence with theories of random
disorder. Thus, the properties of smectic elastomers with a rigid crosslinker and low-
molecular-mass smectics confined within aerosils are rather close to each other.
Nevertheless, the quantitative interpretation of the behavior of smectic networks
constitutes a major theoretical challenge.
The description of crosslinking in smectic elastomers involves effects arising from
internal nonuniform strain. In smectic elastomers prepared according to the two-step
crosslinking process, mechanical strain is imprinted in the system during the uniaxial
alignment. In smectic networks, crosslinks can generate various types of defect with
the associated elastic fields leading to additional stress. Strain-induced broadening of
X-ray peaks is well known in various fields, for example, in metals subjected to cold
work and in certain semiconductors [ 121 , 152 , 153 ] . Generally, two effects contribute
to X-ray peak broadening: the finite size of the crystalline or smectic domains (as
discussed above) and nonuniform strain. The strain broadening of a diffraction peak
leads approximately to a linear increase of Dq z with harmonic number n , whereas the
size effect does not depend on it. Hence, the measured FWHM can be written as in
[ 154 ] , dropping for convenience the index z :

Dq exp ¼ Dq size þ n 2 Dq 2

e ;

(17)

in which Dq e is the strain-induced contribution and the instrumental resolution has
been disregarded for convenience. Experimentally, the width of the quasi-Bragg
peaks along q z increases about linearly with n in agreement with ( 17 ) . In a simple
harmonic description, the width of the smectic peaks would be the same for all
different orders of diffraction. We can in principle separate the two contributions
using ( 17 ) and obtain an average domain size. However, to obtain a good accuracy
several harmonics are needed that are not available for the present elastomers. For
that reason, we have attributed the full width of the first-order peaks to finite-size
effects, which is only approximately correct.

5.2.3 Smectic-A-Nematic Transition

An interesting extension of the above results is obtained if, in the siloxane system
of Fig. 11 , the side group R1 is replaced by a nematogenic side group M3.